a comparison of recent methods for the analysis of small-sample cross-over studies

STATISTICS IN MEDICINEStatist. Med. 2003; 22:2821–2833 (DOI: 10.1002/sim.1537)

A comparison of recent methods for the analysis ofsmall-sample cross-over studies

Xun Chen∗;† and Lynn Wei

Clinical Biostatistics; Merck Research Laboratories; RY34-A316; Rahway; NJ 07065; U.S.A.

SUMMARY

The standard analysis of variance (ANOVA) method is usually applied to analyse continuous datafrom cross-over studies. The method, however, has been known to be not robust for general variance–covariance structure. The simple empirical generalized least squares (EGLS) method, proposed in anattempt to improve the precision of the standard ANOVA method for general variance–covariancestructure, is usually insu�cient for small-sample cross-over trials. In this paper we compare the followingcommonly used or recent approaches: standard ANOVA; simple EGLS; modi�ed ANOVA methodderived from a modi�ed approximate F-distribution; and a modi�ed EGLS method adjusted by theKenward and Roger procedure in terms of robustness and power while applying to small-sample cross-over studies (say, the sample size is less than 40) over a variety of variance–covariance structures bysimulation. We �nd that the unconditional modi�ed ANOVA method has robust performance for all ofthe simulated small-sample cross-over studies over the various variance–covariance structures, and hascomparable power with the standard ANOVA method whenever they are comparable in type I errorrate. The EGLS method (simple or modi�ed) is not reliable when the sample size of a cross-overstudy is too small, say, less than 24 in the simulation, unless a simple covariance structure is correctlyassumed. Given a relatively larger sample size, the modi�ed EGLS method, assuming an unstructuredcovariance matrix, demonstrates robust performance over the various variance–covariance structures inthe simulation and provides more powerful tests than those of the modi�ed (or standard) ANOVAmethod. Copyright ? 2003 John Wiley & Sons, Ltd.

KEY WORDS: ANOVA; cross-over study; EGLS; empirical power; robust; small-sample; variance–covariance structure

1. INTRODUCTION

Cross-over designs have been used extensively in clinical trials for a variety of experimentsincluding phase I/II pharmacokinetic and pharmacodynamic trials and all phases of trials insome chronic diseases. Many cross-over trials, especially those for early phase clinical studies,are small. The analysis methods applied well for large-sample cross-over studies may becomeproblematic when sample size is small. In this paper we will investigate the performance ofdi�erent conventional methods and modi�ed methods for the analysis of small-sample cross-over studies.

∗Correspondence to: Xun Chen, Clinical Biostatistics, Merck Research Laboratories, RY34-A316, Rahway, NJ 07065,U.S.A.

†E-mail: Xun [email protected] December 2001

Copyright ? 2003 John Wiley & Sons, Ltd. Accepted February 2003

2822 X. CHEN AND L. WEI

Suppose we have a s-sequence, p-period cross-over trial comparing t treatments. Assumethere are ni subjects within sequence i (i=1; : : : ; s) and

∑si=1ni= n. Let yijk represent the

response of subject ij (the jth subject within sequence i) at its kth period. The usual linearmodel for this cross-over trial can be written as

yijk =�+ �k + �d[i; k] + �d[i; k−1] + sij + �ijk (1)

where �, �k , �d[i; k] and �d[i; k−1] represent the overall mean, period e�ect for period k, treatmente�ect for the treatment applied in sequence i at period k, and carry-over e�ect from the treat-ment applied at the proceeding period k − 1 in the same sequence i (�d[i;0]≡ 0), respectively,sij represents the subject e�ect for subject j in sequence i and �ijk represents the randomexperimental error for subject j in sequence i at period k. In this paper we limit our interestin the �rst-order carry-over e�ect, that is, assume the carry-over lasts for a single period andis independent of the treatment administered in the period in which the carry-over occurs[1]. A cross-over model only considering the �rst-order carry-over may not be an appropriateor practical model for many situations as pointed out by Senn [2]. It can still be, however,a useful model in some situations. For example, in a cross-over trial including placebo anddrugs A and B, if a previous study shows no drug interaction between drugs A and B, it isthen plausible to assume that the carry-over e�ect from drug A to drug B is similar to thatfrom drug A to placebo. It is, therefore, important to develop ‘correct’ analysis for model (1)for both theoretical and practical needs [3, 4].In this paper we assume the response variable yijk is continuous and normally distributed.

Assume general covariance structure for the random error of each subject, say, var(�ij)=�,and assume random subject e�ect, say, sij ∼ iid N(0; �2s ), and sij is independent of �ij forall i=1; : : : ; s and j=1; : : : ; ni. The covariance structure of the response variable is thusvar(yij)=�=�2s Jp + �, where Jp denotes a p×p dimension matrix of ones. Let � denotethe �xed e�ect parameters (the overall mean, the period, treatment and carry-over e�ects) inthe model (1); the generalized least square (GLS) estimator of � can be represented as

ˆ�GLS = (X′(�−1 ⊗ In)X )−X ′(�−1 ⊗ In)y (2)

with variance

var( ˆ�GLS)= (X′(�−1 ⊗ In)X )− (3)

where y=(y111; : : : ; y11p; : : : ; ysnsp)′, X = (1|X�|X�|X�) represents the design matrix of model

(1) and (X ′�−1X )− denotes the general inverse of matrix X ′�−1X . If � is known, the GLSmethod would be fully e�cient and would be the appropriate one to use.However, � is rarely (if ever) known in practice. The standard analysis of variance

(ANOVA) method estimates the �xed e�ect parameter � by eliminating the subject ef-fect parameters at �rst and assuming �=�2e Ip, which is equivalent to substituting �

−1 byD=�−2e (Ip − 1

pJp) in (2) and (3) and deriving the so-called ordinary least squares (OLS)estimator. The standard ANOVA method is conventionally used to analyse model (1) in prac-tice. It is known that the standard ANOVA method is valid if � obeys a so-called type Hstructure studied by Huynh and Feldt [5]. The assumption of type H structure for the co-variance matrix is not always realistic, however. For example, the example 6.2 in Jones andKenward [3] which referred to a comparison of three drugs A, B and C in their e�ect on

Copyright ? 2003 John Wiley & Sons, Ltd. Statist. Med. 2003; 22:2821–2833

ANALYSIS OF SMALL-SAMPLE CROSS-OVER STUDIES 2823

systolic blood pressure through a three-period six-sequence cross-over trial corresponds to anobserved covariance matrix like

111:75 99:00 91:42

99:00 103:67 118:75

91:42 118:75 228:50

which obviously departed from the type H structure. To make the standard ANOVA valid forgeneral covariance structure, Bellavance et al. [6] suggested modifying the F-test statisticsof standard ANOVA and applying an adjusted approximate F-distribution for the relevanthypothesis tests (referred to as modi�ed ANOVA method hereafter). The simulation studiesby Bellavance et al. showed that the modi�ed ANOVA method gave a very good control onthe type I error rate over a variety of structures of � for three-period cross-over studies withsmall to moderate sample sizes.Rather than assuming an unveri�ed uniform covariance matrix, a practical alternative is an

empirical form of the GLS method which substitutes � by an estimate of it, namely, theEGLS method [3]. Although the EGLS method is proposed in an attempt to improve theprecision of the OLS based estimator, there is no guarantee that the EGLS method will bemore precise than the OLS method, especially in small-sample cases when the precision in theestimate of � can have signi�cant impact on the precision of the EGLS estimator, say, �EGLS.To reduce the bias and impact due to small-sample size, Kenward and Roger [7] suggesteda procedure adjusting the conventional EGLS asymptotic inference, which is generally moreappropriate for a large sample study. We will refer to the EGLS method with the Kenwardand Roger adjustment as the modi�ed EGLS method and the plain EGLS method without theKenward and Roger adjustment as the simple EGLS method hereafter.At this time it is unclear as to which method – the modi�ed ANOVA method or the modi�ed

EGLS method – should be adopted to replace the standard ANOVA method to provide robustanalyses for small-sample cross-over studies. The practitioners may �nd it di�cult to make achoice from the di�erent methods, either at design stage or analysis stage, and as shown inSection 3, applying these di�erent methods may result in quite di�erent results.To provide a practical guidance for the analysis of a small-sample cross-over study speci�ed

as model (1), we conduct a Monte Carlo study to compare the performance of the standardANOVA method, the simple EGLS method, the modi�ed ANOVA method and the modi�edEGLS method with regard to their type I error rates for various variance–covariance structuresin small three- and four-period cross-over studies. The power of the four di�erent methodsare compared whenever they are comparable in type I error rates.The next section brie�y introduces the modi�ed ANOVA method and the modi�ed EGLS

method. These methods are compared by Monte Carlo simulations in Section 3.

2. METHODS

2.1. Modi�ed ANOVA method

As in the standard ANOVA method, the modi�ed ANOVA method applies the OLS method toestimate the �xed e�ect parameter � in model (1). The inference of the estimates is, however,



properly adjusted in the modi�ed ANOVA method compared with that in the standard methodto address the possible departure of � from the type H structure.Consider the two hypotheses, H0� : �1 = · · · = �t and H0� : �1 = · · · = �t , that we are inter-

ested in most of the time. In the standard ANOVA method, the F-ratios of the hypothesistests are de�ned as

f=MSRMSE

where MSE= y′EyRank(E) , E= I − X (X ′(D ⊗ In)X )−X ′(D ⊗ In), MSR=

y′RyRank(R) , R=X (X

′(D ⊗In)X )−X ′(D⊗ In)−M (M ′(D⊗ In)M)−M ′(D⊗ In), M = (1|X�|X�) when testing the hypothesisof no treatment e�ects (H0�), and M = (1|X�|X�) when testing the hypothesis of no carry-overe�ects (H0�) [6]. The standard ANOVA method conducts an F-test following the distributionassumption

f ∼ F(Rank(R);Rank(E))under the corresponding null hypothesis. The distribution assumption is not valid if the co-variance matrix structure is non-type H [8]. For general structure of �, Bellavance et al. [6]recommended modifying the above F-test by assuming f ∼ bF(d1; d2), with

b=Rank(E)Tr[R(�⊗ In)]Rank(R)Tr[E(�⊗ In)]

d1 =Tr2[R(�⊗ In)]Tr[R(�⊗ In)]2

d2 =Tr2[E(�⊗ In)]Tr[E(�⊗ In)]2

The approximation reduces to the standard F-distribution of the standard ANOVA methodwhen � is compound symmetry or the more general type H structure. Without any priorknowledge on the structure of �, one may estimate it by the simple within-sequence sampledispersion matrix, say

S=1

n− ss∑i=1

ni∑j=1(yij − �yi)(yij − �yi)

′

where �yi=1ni

∑nii=1yij; i=1; : : : ; s. This simple estimation was found generally su�cient for

the purpose of modi�cation [6]. Robust interval estimates for any linear contrast of treatmentor carry-over e�ects under general � structures can also be derived based on the modi�edF-test [8].The modi�ed ANOVA method can be implemented either conditionally or unconditionally.

The conditional modi�ed ANOVA method �rst tests the structure of the variance–covariancematrix of y (for example, Mauchly’s sphericity test [9]) then conducts modi�cation if signif-icant departure from the type H structure is detected, or uses the standard ANOVA methodotherwise. The unconditional modi�ed ANOVA method, on the other hand, uniformly adoptsthe modi�ed procedure independent of the variance–covariance structure of y – type H ornot.



2.2. Modi�ed EGLS method

The empirical generalized least squares (EGLS) method estimates the parameter � by substi-tuting the unknown � with its estimate, that is

ˆ�EGLS = (X′�

−1X )−X ′�

−1y (4)

where � denotes an estimate of �. A simple estimator for � is the within-sequence sampledispersion matrix S shown in the earlier section. Re�ned estimates can be derived using themaximum likelihood (ML) method or the restricted maximum likelihood (REML) method.Iterations are generally needed when applying these two methods and the within-sequencesample dispersion matrix S is usually used as the initial estimate. The simulation evidencepresented by Swallow and Monahan [10] favoured the REML method over other covarianceestimators. We will use the REML method to estimate the covariance parameters in this paper.The F-test statistic for any linear contrast of the period, treatment or carry-over e�ect, say

for H0 :L′�=0, in the EGLS type method is conventionally set as

f=ˆ�′L′(L var( ˆ�)L′)−1L ˆ�

Rank(L)(5)

The asymptotic-based estimate for var( ˆ�) is conventionally set as (X ′�−1X )−, which takes

no account for the variability of � and tends to seriously underestimate the true sampling

variability of ˆ� when sample size is small and signi�cantly impact the corresponding hy-

pothesis test and con�dence interval estimate. The internal random structure of X ′�−1X also

complicates the inference as it needs to be taken into account when deriving the denominatordegree of freedom for the F-test statistic, especially when sample size is small.Kenward and Roger [7] proposed a procedure to provide more appropriate small-sample

inference for the EGLS method. The Kenward and Roger method �rst in�ated the conventional

estimate for var( ˆ�) by adding in a term to it. It then substituted the adjusted variance estimateinto (5) and estimated the denominator degree of freedom for the F-test statistic based onit. It is believed that the EGLS method with inference developed based on the Kenward andRoger procedure is generally more reliable than the simple EGLS method for small trials.However, at the moment, it is still unknown how the modi�ed EGLS method compares withthe modi�ed ANOVA method and how small the sample size can go before the modi�edEGLS method collapses.The EGLS method may be easily conducted using a SASTM procedure – SASTM Proc Mixed.

The Kenward and Roger adjustment became available in SASTM Proc Mixed after the SASTM

2000 version [11], which may be called by specifying the denominator degree of freedomoption (DDFM) as KENWARDROGER. Besides the Kenward and Roger method, SASTM

Proc Mixed provides �ve di�erent ways to estimate the denominator degree of freedom of (5)including the well known and often used Satterthwaite approximation [12]. SASTM Proc Mixedallows the user to specify a structure pattern for �. (SASTM Proc Mixed provides at least 20options, including an unstructured choice, in its recent version.) There are di�erent ways tospecify a same covariance structure in SASTM Proc Mixed, and most of the time, it is su�cient(and also recommended) to de�ne the variance–covariance matrices of the random e�ect



term and the random error term jointly through the REPEATED statement [13]. Generallymore powerful analysis can be achieved if the structure is speci�ed correctly, but as a trade-o�, one also needs to take the risk of misspecifying the variance–covariance pattern. Thesimulation study by Wright and Wol�nger [14] showed how the type I error could be in�atedwhen a wrong variance–covariance structure was �tted. In this paper, we avoid the risk ofmisspecifying variance–covariance structure by simply assuming an unstructured pattern, andfocus our investigation on the e�ect of small-sample size. The potential loss of power due tothe speci�cation of an unstructured pattern is evaluated by the simulation study presented later.

3. MONTE CARLO SIMULATION AND PERFORMANCE EVALUATION

Simulations are conducted through the SASTM software package to compare the standardANOVA method (SASTM Proc GLM), the modi�ed ANOVA method (coded with SASTM ProcIML), the simple EGLS method and the modi�ed EGLS method (SASTM Proc Mixed). (Thecode is available from the �rst author upon request.) The response variable y is generated intwo steps. At �rst, a normal random vector with a pre-assigned variance–covariance matrix of�⊗In and mean zero is generated; the random vector is then added to X (�; �1; : : : ; �p; �1; : : : ; �t ;�1; : : : ; �t)′ with prespeci�ed values for �, �=(�1; : : : ; �p)′, �=(�1; : : : ; �t)′ and �=(�1; : : : ; �t)′.Our interest focuses on the test for treatment and carry-over e�ects. Without loss of generality,we assume �=�k =0 for all k throughout the simulation study.Let N0 denote the number of independent cross-over trials generated at a null hypothesis and

N1 be the number of independent cross-over trials generated at an alternative in simulation.Let T0i denote the test statistic of the ith simulated trial under the null hypothesis and T1idenote the test statistic of the ith simulated trial under the alternative hypothesis. Let C�denote the critical value of nominal level �. The empirical type I error rate of the test isestimated by 1

N0

∑N0i=1I(T0i¿C�), where I(A)=1 if A is true and 0 otherwise. The empirical

power at the alternative is estimated by 1N1

∑N1i=1I(T1i¿C�). To make the power comparison

more meaningful, the adjusted empirical power is derived following 1N1

∑N1i=1I(T1i¿C�), where

C� denotes the (1 − �) quantile of the null test statistics, that is, C�= the (1 − �)N0th valueof T01; : : : ; T0N00 when these are placed in order from smallest to largest [15]. As a result,the powers of di�erent tests are compared based on the same empirical signi�cance level. Inthis paper, we choose N0 =N1 = 5000 and we only present the results for the nominal level�=0:05. Results for the other nominal levels are similar and are available from the �rst authorupon request. The standard error of the empirical type I error rates, say �, can be estimatedby

√{�(1− �)=N0}.

3.1. Three-period cross-over design

We �rst test the performance of the four di�erent methods in a balanced three-period cross-over design, which compares three di�erent treatments, say, A, B and C, and is formed bysix di�erent sequences, ABC, BCA, CAB, ACB, BAC and CBA. The sample size of thecross-over trial varies from 12 to 36, corresponding to two subjects per sequence to sixsubjects per sequence. (We also considered the situations when the number of subjects ineach sequence is slightly unbalanced. Similar simulation results are observed and are not



Table I. Variance–covariance matrices (V ) used in the three-period cross-over simulations.

Compound symmetry Type H No structure (1) No structure (2) No structure (3)

1.00 0.50 0.50 0.333 0.167 0.500 0.242 0.242 0.135 1.131 0.905 0.377 1.235 1.092 1.0040.50 1.00 0.50 0.167 1.00 0.833 0.242 0.967 0.541 0.905 1.131 0.611 1.092 1.147 1.3130.50 0.50 1.00 0.500 0.833 1.667 0.135 0.541 1.209 0.377 0.611 1.131 1.004 1.313 2.526

reported here.) Assuming the observations from the n di�erent subjects of a cross-over studyare mutually independent and identically distributed, we have var(y)=�⊗ In. Table I showsthe �ve di�erent variance–covariance matrices tested in the simulation. The �rst one representsthe compound symmetry structure, the second one represents the type-H structure, and thelast three have no speci�c structure. The �rst four covariance matrices are all arbitrarilychosen, whereas the last one is a standardized version of the empirical covariance matrix fromthe Jones and Kenward [3] cross-over trial example shown in Section 1. All of these �vedi�erent variance–covariance matrices are standarized to share the same value of

∑3i=1�ii −∑3

i; j=1i¿j

�ij=1:5, which implies that the simulated three-period cross-over trials have the same

average variance for the di�erence of observations from di�erent periods, that is

13 (var(yij1 − yij2) + var(yij1 − yij3) + var(yij2 − yij3))≡ 1:0

3.1.1. Type I error under the null hypotheses. Under the null hypotheses of no treatmentand no carry-over e�ects, we have �= �=0. Table II summarizes the empirical type I errorrates of the four di�erent approaches under the �ve di�erent types of variance–covariancematrices, where ANOVA std represents the standard ANOVA method; ANOVA mdf(cond.)represents the conditional modi�ed ANOVA method and ANOVA mdf(uncond.) the uncon-ditional modi�ed ANOVA method. When applying SASTM Proc Mixed to generate the EGLStype estimator and corresponding inference, the �ve di�erent types of denominator degreeof freedom estimation methods now available in the procedure are tested. Only the resultscorresponding to the default estimation method and the Kenward–Roger estimation methodare reported in Table II (denoted as EGLS smpl and EGLS mdf, respectively). The otherthree estimation methods (including the Satterthwaite method) have similar performance tothe default one. The variance–covariance matrix may be speci�ed as an unstructured one(UN in parentheses), or as a type H matrix (HF in parentheses), or as the default compoundsymmetry matrix (CS in parentheses).The simulation results indicate that:

(i) The standard ANOVA method performs well under the compound symmetry and typeH variance–covariance structure, as it should be, yielding a type I error rate near thenominal level 5 per cent. However, under the three variance–covariance matrices withno structure, the standard ANOVA method tends to have an in�ated type I error ratethat does not improve as sample size increases.

(ii) The conditional modi�ed ANOVA method has a similar performance as the standardANOVA method under the compound symmetry structure and type H structure, andhas better controlled type I error rate under the three unstructured variance–covariancematrices as sample size increases.



Table II. Type I errors for the analyses of three-period cross-over trials (based on 5000 simulations,nominal level �=5 per cent).

ni =2; n=12 ni =3; n=18 ni =4; n=24 ni =5; n=30 ni =6; n=36

H0� H0� H0� H0� H0� H0� H0� H0� H0� H0�

Compound symmetryANOVA Std 5.0 5.4 5.3 5.0 5.3 4.9 4.6 5.0 5.6 4.8ANOVA Mdf(cond.) 5.0 5.8 5.3 5.2 5.4 5.2 4.5 5.1 5.5 4.9ANOVA Mdf(uncond.) 5.0 7.5 5.3 6.1 5.4 5.9 4.5 5.9 5.4 5.4EGLS Smpl (CS) 4.8 5.8 5.3 5.0 5.4 5.5 4.6 5.4 5.7 4.9EGLS Mdf (CS) 4.8 5.6 5.5 5.8 5.4 5.5 4.6 5.3 5.7 4.9EGLS Smpl (UN) 21.9 21.4 12.4 11.6 9.4 9.3 7.5 7.4 7.6 6.7EGLS Mdf (UN) 10.0 9.4 7.1 6.5 6.3 5.9 5.0 5.3 5.3 4.8

Type HANOVA Std 5.4 5.2 5.1 4.7 5.4 5.3 4.5 5.2 4.8 4.4ANOVA Mdf(cond.) 5.5 5.6 5.1 4.9 5.5 5.4 4.5 5.3 4.7 4.5ANOVA Mdf(uncond.) 5.7 7.4 4.8 6.2 4.9 6.0 4.4 5.8 4.7 5.0EGLS Smpl (CS) 4.4 5.0 3.8 4.6 3.7 5.3 3.3 5.4 3.6 4.3EGLS Mdf (CS) 4.4 5.0 3.7 4.5 3.7 5.2 3.3 5.3 3.6 4.3EGLS Smpl (UN) 21.3 19.6 11.6 11.6 9.2 8.5 7.7 7.3 6.6 6.2EGLS Mdf (UN) 9.8 7.7 6.4 6.0 5.9 5.5 4.8 4.9 5.2 4.5

No structure (1)ANOVA Std 7.4 8.0 6.9 7.9 7.1 7.7 6.7 7.5 6.6 8.1ANOVA Mdf(cond.) 7.2 7.7 6.6 7.2 6.6 7.1 6.1 6.9 6.0 7.2ANOVA Mdf(uncond.) 5.7 7.1 5.6 5.9 5.4 5.5 5.0 5.5 5.0 5.7EGLS Smpl (CS) 6.4 8.8 5.9 8.5 6.2 8.4 5.5 8.5 5.6 8.9EGLS Mdf (CS) 6.3 8.5 5.9 8.3 6.2 8.2 5.5 8.3 5.6 8.8EGLS Smpl (UN) 17.5 16.7 10.4 9.7 8.1 7.8 7.5 6.5 6.9 7.0EGLS Mdf (UN) 8.7 8.0 6.5 5.8 5.7 5.3 5.4 4.7 5.3 5.5



(iii) The unconditional modi�ed ANOVA method has consistently better performance thanthe conditional one under the three unstructured variance–covariance matrices – theperformance of the latter method primarily depends on the power of the tests for the



type H structure, which is usually very poor in small-sample cases [16]. Under thecompound symmetry structure and the type H structure, when no correction is neededtheoretically, the unconditional modi�ed ANOVA method performs comparably withthe conditional method and the standard ANOVA method.

(iv) The EGLS method with variance–covariance structure speci�ed as compound sym-metry, that is, TYPE=CS in SASTM Proc Mixed, has similar performance using thedefault degree of freedom estimation method or the KR estimation method. It performswell if the true variance–covariance structure is compound symmetry, tends to have atype I error rate smaller than the nominal level for the treatment e�ect test if the truevariance–covariance structure is type H, and has an noticeably in�ated type I errorrate for the three unstructured variance–covariance matrices.

(v) The simple EGLS with variance–covariance matrix speci�ed as unstructured performspoorly under all variance covariance structures for small n’s. The performance im-proves slowly as sample size increases. (At the largest sample size of 36 in thesimulation, the type I error rate is still at 6.5–8.5 per cent level for the nominal levelof 5 per cent.)

(vi) The modi�ed EGLS method assuming unstructured variance–covariance matrix hada larger than nominal type I error rate when sample size is small, say n¡24. Theperformance improves rapidly as sample size increases.

In summary, the unconditional modi�ed ANOVA method demonstrates robust performanceacross di�erent variance–covariance structures and di�erent sample sizes for the tested three-period cross-over design. When sample size is larger than or equal to 24, the modi�ed EGLSmethod assuming unstructured variance–covariance matrix has a comparable performance tothe unconditional modi�ed ANOVA method in terms of type I error rate in the simulationstudy.

3.1.2. Power under selected alternatives. Now focusing on the various versions of the fourdi�erent methods which have comparable type I error rates in Table II, we compare their‘adjusted’ empirical power [15] under a selected alternative, say �=(0; 0; 0:6), �=(0; 0; 0:6),in Table III. We have the following observations:

(i) The power of the unconditional modi�ed ANOVA method is similar to that of the stan-dard ANOVA method when the latter is valid, that is, when the variance–covariancematrix is compound symmetry or type H.

(ii) The power of modi�ed EGLS method assuming unstructured variance–covariance ma-trix is only slightly reduced compared with a speci�cation that agrees to the truevariance–covariance structure.

(iii) Except when the true variance–covariance matrix is compound symmetry, the modi�edEGLS method assuming an unstructured variance–covariance matrix provides morepowerful tests than the unconditional modi�ed ANOVA approach.

3.2. Four-period cross-over design

The four-period cross-over design we test here is the conventional four-period William design:comparing four di�erent treatments, say, A, B, C and D, in four di�erent sequences, ABDC,BCAD, CDBA and DACB. The sample size of the cross-over trial varies from 8 to 32,



Table III. Adjusted empirical power of tests for three-period cross-over trials (based on 5000 simulations,nominal level �=5 per cent).

ni =4; n=24 ni =5; n=30 ni =6; n=36

H0� H0� H0� H0� H0� H0�

Compound symmetry ANOVA Std 74.0 49.3 85.7 57.6 89.9 68.8ANOVA Mdf(uncond.) 73.9 45.9 85.8 55.5 90.2 66.0EGLS Mdf (CS) 74.8 50.4 86.0 60.5 90.3 70.7EGLS Mdf (UN) 65.6 43.4 81.8 55.3 88.2 67.0

Type H ANOVA Std 75.2 49.9 87.2 59.0 91.8 69.3ANOVA Mdf(uncond.) 75.3 45.5 86.5 57.3 91.7 67.3EGLS Mdf (HF) 98.0 69.8 99.7 83.4 99.9 90.5EGLS Mdf (UN) 97.0 66.2 99.5 81.2 99.9 89.6

No structure (1) ANOVA Mdf(uncond.) 68.9 39.6 81.1 49.6 88.4 57.0EGLS Mdf (UN) 86.8 39.8 95.0 54.1 98.2 59.6



Table IV. Variance–covariance matrices (V ) used in the four-period cross-over simulations.

Compound symmetry No structure

1.00 0.50 0.50 0.50 0.94 0.75 0.31 0.160.50 1.00 0.50 0.50 0.75 0.94 0.51 0.250.50 0.50 1.00 0.50 0.31 0.51 0.94 0.660.50 0.50 0.50 1.00 0.16 0.25 0.66 0.94

corresponding to two subjects per sequence to eight subjects per sequence. (Similar simulationresults are observed when the number of subjects in each sequence is slightly unbalanced.)Again let var(y)=�⊗ In. Table IV presents two di�erent variance–covariance matrices usedin the simulation. The �rst one represents the compound symmetry structure and the secondone has no speci�c structure. These two di�erent variance–covariance matrices have a constantvalue of 3

∑4i=1�ii − 2

∑4i; j=1i¿j

�ij=6, which implies the corresponding four-period cross-over

trials have the same average variance for the di�erence of observations from di�erent periods,that is,

16 (var(yij1 − yij2) + var(yij1 − yij3) + var(yij1 − yij4) + var(yij2 − yij3)+var(yij2 − yij4) + var(yij3 − yij4))≡ 1:0

3.2.1. Type I error under the null hypotheses. Table V summarizes the empirical type I errorsof the four di�erent methods under the two di�erent types of variance–covariance matrices.



Table V. Type I errors for the analyses of four-period cross-over trials (based on 5000 simulations,nominal level �=5 per cent).

ni =2; n=8 ni =4; n=16 ni =6; n=24 ni =8; n=32

H0� H0� H0� H0� H0� H0� H0� H0�

Compound symmetryANOVA Std 5.0 5.0 5.2 4.5 5.5 4.9 5.0 4.8ANOVA Mdf(cond.) 4.8 5.0 5.2 4.5 5.5 4.9 5.1 4.8ANOVA Mdf(uncond.) 3.4 4.9 4.9 5.3 5.2 5.0 5.1 5.2EGLS Smpl (CS) 5.0 5.1 5.3 4.4 5.7 5.0 5.2 4.7EGLS Mdf (CS) 5.0 5.1 5.3 4.3 5.7 5.0 5.2 4.6EGLS Smpl (UN) 50.4 50.7 13.8 13.6 9.3 9.2 8.1 7.7EGLS Mdf (UN) 19.7 20.6 7.3 6.4 6.2 6.0 6.0 5.3

No structureANOVA Std 11.8 10.4 10.8 9.7 10.7 9.8 10.4 9.5ANOVA Mdf(cond.) 10.7 9.9 7.1 6.3 4.8 5.2 5.0 5.2ANOVA Mdf(uncond.) 4.7 5.7 5.4 5.2 4.6 4.9 5.0 5.2EGLS Smpl (CS) 11.9 10.6 11.0 9.9 10.6 10.1 10.5 9.1EGLS Mdf (CS) 11.8 10.4 11.0 9.9 10.5 10.0 10.4 9.1EGLS Smpl (UN) 49.6 53.9 12.5 13.6 8.8 9.3 6.8 7.9EGLS Mdf (UN) 20.3 22.6 6.4 7.0 5.6 5.8 5.1 5.5

Table VI. Adjusted empirical power of tests for four-period cross-over trials (based on 5000 simulations,nominal level �=5 per cent).

ni =6; n=24 ni =8; n=32

H0� H0� H0� H0�

Compound symmetry ANOVA Std 79.3 64.1 91.4 77.1ANOVA Mdf(uncond.) 79.3 62.8 91.1 76.1EGLS Mdf (CS) 79.3 64.5 91.4 78.2EGLS Mdf (UN) 72.5 55.7 87.5 72.3

No structure ANOVA Mdf(uncond.) 66.3 49.0 80.6 65.6EGLS Mdf (UN) 82.9 61.80 94.3 80.8

The simulation results for the four-period cross-over design are similar to those for the three-period cross-over design. In general, the unconditional modi�ed ANOVA method has a type Ierror rate close to the nominal level for the two di�erent variance–covariance matrices andthe di�erent sample sizes. When sample size is larger than or equal to 24, the modi�ed EGLSassuming unstructured variance–covariance matrix has a comparable type I error rate as theunconditional modi�ed ANOVA method.

3.2.2. Power under selected alternatives. Table VI compares the adjusted empirical powerof the di�erent methods under an alternative �=(0; 0; 0; 0:6), �=(0; 0; 0; 0:6) in the situationswhen the type I error rates of the di�erent methods are comparable. Similar to the three-periodcross-over study, the modi�ed EGLS method assuming unstructured variance–covariance



matrix provides a more powerful test than the unconditional modi�ed ANOVA method (whenthese two methods have comparable type I error rates which are close to the nominal level).

4. CONCLUSIONS AND DISCUSSION

The unconditional modi�ed ANOVA method provides very good control on the type I errorrate of small-sample cross-over studies over a variety of variance–covariance structures inthe simulation study. It outperforms the conditional modi�ed ANOVA method in most ofthe time due to the low power of the Mauchley’s sphericity’s test in small-sample cases.The standard ANOVA method can provide valid analysis for a cross-over study with type Hvariance–covariance matrix, but verifying the structure of the variance–covariance matrix isnot straightforward in practice and usually has low power in small-sample cases. Moreover,we �nd the empirical power of the standard ANOVA method is very close to that of the un-conditional modi�ed ANOVA method in situations where they are comparable in type I errorrates. The performance of the EGLS method is sensitive to the setting of variance–covariancestructure and the way of estimating the denominator degree of freedom. The unstructuredvariance–covariance matrix (SASTM Proc Mixed with TYPE=UN) helps the EGLS methodto achieve robust tests for a variety of variance–covariance structures. The modi�ed EGLSmethod (the one with Kenward–Roger adjustment) noticeably surpasses the simple EGLSmethods in the simulated small-sample cross-over studies. The EGLS method, modi�ed orunmodi�ed, is generally not reliable when the sample size of a cross-over study is too small,say, less than 24, as shown in the simulation, unless a simple covariance structure is cor-rectly assumed. Given relatively larger sample size, the modi�ed EGLS method assumingunstructured variance–covariance matrix demonstrates robust performance over the variousvariance–covariance structures, and is usually more powerful than the modi�ed (or standard)ANOVA method in situations where they have comparable type I error rates. In conclusion, werecommend adopting the unconditional modi�ed ANOVA method for the cross-over studiesuniformly when sample size is less than 24, and apply the modi�ed EGLS method, assumingunstructured variance–covariance matrix, for larger cross-over studies.Throughout the paper we assumed random subject e�ect. What should be noted is that

applying a �xed subject e�ect model or a random subject e�ect model should primarilydepend on which population the results are inferred to. When �xed subject e�ect is assumed,the ANOVA type method (the standard ANOVA or the modi�ed ANOVA) performs thesame as in the random subject e�ect case as the ANOVA type method is conducted througheliminating the subject e�ect terms from the model prior to estimating the other interestedparameters. The EGLS method conducted through SASTM Proc Mixed will, however, performless powerfully when �xed subject e�ect is assumed as more nuisance parameters (one foreach subject) are involved. Its power advantage against the ANOVA type method diminishes.Finally, as we commented in Section 1, for some cross-over trials, the model with carry-over

e�ect is not plausible due to a long washout period or the knowledge on subject matter anda simpler model without carry-over e�ect is preferred [2]. The simpli�ed model, containingonly treatment, period and subject, has many advantages when compared with model (1): thetreatment e�ect is best estimated by a simple formula without involving correction for thecarry-over e�ect; the standard ANOVA can be more robust under various variance–covariancestructures other than type H; non-parametric methods, including the general permutation test



(for example, Ohrvik [17]), are easier to derive etc. More on the simpli�ed model will befurther explored in a future study.

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a comparison of recent methods for the analysis of small-sample cross-over studies

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